Numerical Model of the Insertion Loss Promoted by the Enclosure of a Sound Source

Gil F. Greco*1, Bernardo H. Murta1, Iam H. Souza1, Tiago B. Romero1, Paulo H. Mareze1, Arcanjo Lenzi2

and Júlio A. Cordioli2.

1. Undergraduate Program in Acoustical Engineering, Federal University of Santa Maria, RS, Brazil

2. Laboratory of Vibrations and Acoustics, Federal University of Santa Catarina, SC, Brazil

*Corresponding author: Av. Roraima 1000, Santa Maria, RS, Brasil. 97105-900, gil.greco@eac.ufsm.br

Abstract: Usually, the enclosure of a sound

source is employed in order to control the noise

radiated by industrial machines. This structure

changes the path of sound transmission between

the sound source and the receiver, imposing a

high impedance to the wave propagation.

However, the enclosure design requires attention

since the enclosure and its panel’s vibrational

modes will influence its performance.

Furthermore, the enclosure’s structure response

and the acoustic field must be correctly coupled.

Although there are models to assess the acoustic

performance of enclosures, the multiphysics

nature of the problem makes its analytical

modeling unfeasible in practice. Thus, this study

aims to develop and validate a finite element

numerical model to represent the Insertion Loss

(IL) promoted by the enclosure of a sound source.

For the validation, a enclosure prototype was

built in wood and the IL was measured in

laboratory. The idea is to develop an efficient

numerical model that would be suitable for

enclosure's design and optimization.

Key-words: Noise Control, Enclosure Design,

Acoustic-Structure Interaction, Vibroacoustics.

1. Introduction

Usually, it is desirable to control the sound

radiation of a certain device, however, most of the

times there is no possibilities to modify the

constructive characteristics of a sound source. In

this cases, enclosing the sound source may be a

convenient way to promote a proper sound

insulation. In fact, Randall [1], states that the use

of a enclosure is the most practical solution when

the required noise level reduction is greater than

10 dB. This approach modifies the sound

transmission path between the sound source and

the receiver, imposing a large impedance to the

propagation of the sound wave [2]. Thus, it can be

used to control the noise radiated by machines or

noisy devices such as compressors, gears and

motors, among others industrial devices.

Although there are models to assess the

acoustic performance of enclosures [3,4], the

multiphysics nature of the problem makes its

analytical modeling unfeasible in practice.

Several aspects must be taken into account when

designing an enclosure. First, the need of a good

accessibility to the sound source can be a

drawback since it requires apertures in the

enclosure, fact that could allow sound leakage

through it. Furthermore, the enclosure's structure

will accommodate acoustic modes in the air

volume within it, which will increase the intensity

of some spectral components of the noise

transmitted by the source. These modes can be

avoided by the optimization of the geometry of

the enclosure as well as through the use of sound

absorbent materials inside the enclosure’s walls.

Another key point is the vibroacoustic

response of the enclosure structure itself. In

certain wavelengths, the structure will

accommodate vibrational modes whose will

reduce substantially the sound insulation in their

resonance frequencies. Taking all these facts in

mind, the correct coupling between the

enclosure’s structural response and the acoustical

field must be done to allow the development of a

representative numerical model of this situation.

This coupled analysis is widely studied and

modeled by several authors [5,6,7] using the

Finite Element (FE) method.

This study goal is to develop and validate a

finite element numerical model to represent the

Insertion Loss (IL) promoted by the enclosure of

a sound source. According to Blanks [2], the

efficiency of an enclosure can be better

represented by the insertion loss (IL), which

quantifies the decrease in the sound power

radiated due to insertion of the enclosure over the

acoustic source. For the validation, one simple

Excerpt from the Proceedings of the 2015 COMSOL Conference in Curitiba

enclosure prototype was built in only one

material, wood (Brazilian ipe) and the sound

power was measured in according to ISO 3741

[8]. Further details about the experimental

characterization of the problem are exposed in

Section 2. The computational models

implementations are fully explained in Section 3.

The results are discussed in Section 4, while

Section 5 presents the conclusions and final

considerations about this study.

2. Experimental Characterization

According to ISO 3741, the measurement of

sound intensity in a surface over the sound source

can be done to derive its sound power level. This

procedure is done by measuring the sound

intensity, emitted by a spherical shaped source

with one speaker playing white noise, in 72

isotropic points positioned in a cubical grid of 1

m³. The measurements were carried in a

reverberant chamber using type-1 hardware. In

order to soften reflections from the chamber’s

walls, 5 square meters of sound absorbing

material was added on the walls. According to the

theory, this measurement can be conducted in

ordinary rooms since it extracts the intensity and

radiated power from the enclosed grid points. The

Sound Power (W) can be derived from the

summation of the normal surface intensity

magnitude, times the surface segment area of the

associated "𝑖" point of the grid:

𝑊 = ∫ 𝐼. 𝑑𝑆 = ∑ 𝐼𝑖 . Δ𝑆𝑖𝑁𝑖=1 . (1)

The frequency dependent Sound Intensity, in

W m2⁄ , can be derived using a 𝑝-𝑝 probe by [9]:

𝐼(𝜔) =−1

𝜔𝜌0Δ𝑟Im[𝑆12(𝜔)], (2)

where 𝜔 is the angular frequency, Δ𝑟 = 12 mm is

the microphone spacing and 𝑆12 is the cross power

spectrum of the pressure signal from the mics 1

and 2. Thus, the Sound Power Level (𝐿𝑤) is given

by:

𝐿𝑤 = 10 log10 (𝑊

𝑊𝑟𝑒𝑓), (3)

where 𝑊𝑟𝑒𝑓 = 10−12 W, is the reference power.

In order to derive the insertion loss of a real

enclosure, a rectangular box (32 cm x 22 cm x

26.5 cm) with 5 fixed sides and opened at the

bottom, was built in a Brazilian species of dry

hardwood. All walls have the same thickness (𝑑 =

24 mm). The same measurement procedure was

repeated with the enclosed source, deriving the

spectral sound power level with the enclosure’s

attenuation. The difference between both results

is the IL of the prototype, and was used to validate

the FE model.

The experimental sound source, shown in

Figure 1, is a spherical speaker with acoustic

frequency response range of 80 Hz to 20 kHz,

electrical impedance of 8 ohm and diameter of 12

cm.

Figure 1. Experimental Sound Source.

The experimental wood enclosure inside of

grid surface during the measurement is in Figure

2.

Figure 2. Sound Power Measurement Using 72

Points of Intensity Data.

3. Computational Model

The problem was modeled using two different

approaches. First, the Pressure Acoustics,

Frequency Domain interface was used to model

the sound source, over a finite circular baffle,

radiating in a free field condition. The second

model includes the enclosure, using the Acoustic-

Shell Interface, Frequency Domain to couple the

structural and acoustic domains. Both 3D models

Excerpt from the Proceedings of the 2015 COMSOL Conference in Curitiba

were set in COMSOL Multiphysics and have in

common the basic geometry, sound source and

mesh discretization.

3.1 General Properties

On the first hand, to model the radiating

source with the absence of the enclosure, two

concentric hemispheres of radius 𝑟1 = 50 cm and

𝑟2 = 51 cm were designed to represent the fluid

domain (air: 𝜌0 = 1.21 kg/m³, 𝑐0 = 343 m/s).

A Spherical Wave Radiation condition is applied

on the surface of the larger hemisphere to emulate

a non-reflective outer boundary condition. The

inner hemisphere surface is used to derive the

pressure values on its surface.

On the second hand, keeping the previous

geometry, a rectangular box (32 cm x 22 cm x

26.5 cm) was centered on the plane surface of the

hemispheres. The structural domain was modeled

using a Shell Structural Element. This way, a

Elastic Material was defined with the following

properties1: 𝐸 = 18 GPa, 𝜌 = 1089.2 kg/m³,

𝜈 = 0.29, and was coupled with the acoustic

domain using the Acoustic-Shell Interaction

interface. All enclosure’s edges were fixed,

except for the four ones in contact with the Hard

Boundary that models the floor.

3.2 Sound Source

The sound source shown in Figure 1 is

modelled as a Power Point Source positioned 2

mm above a sound hard sphere of 6 cm of radius

and its lower extreme was positioned 2 cm above

the floor. This sphere represents the geometry of

the sound source used in the experimental

validation. The source average sound power was

measured in narrow bands according to ISO 3741

[5] and was used as an input to the models.

Although the source sound power is known, it is

important that the numerical model takes into

account the source volume to represent its

influence on the acoustical field. An illustration of

the coupled acoustics-structure model is shown in

Figure 3.

1 Where 𝐸 is the Young’s Modulus, 𝜌 is the

material’s density and 𝜈 is the Poisson’s Ratio.

Figure 3. Coupled Acoustic-Structure Model

Illustration.

3.3 Mesh Discretization

In acoustics, the common approach is to

discretize the domain in terms of elements per

wavelengths. As stated by Atalla [10], at least six

elements per wavelength should be used to avoid

convergence problems. The mesh elements were

dimensioned in order to derive correct results up

to the maximum frequency of analysis, 𝑓𝑚𝑎𝑥 = 2kHz. The element size can be derived by the

simple relation:

ℎ = 𝜆/𝑛 =𝑐0

𝑛𝑓𝑚𝑎𝑥, (4)

where ℎ is the element size, in meters, 𝜆 is the

wavelength and 𝑛 is the desired number of

elements per wavelength. Thus, the mesh was

built with tetrahedral elements, which dimensions

respects the minimal condition to fit at least six

elements per wavelength. The final mesh

discretization was defined with elements of

maximum size ℎ = 2.8 cm.

3.4 Derived Results

As the numerical model validation was

intended to be done by means of IL, it is required

to estimate the Average Sound Power Level (𝐿𝑤).

The sound power can be calculated in terms of

Sound Pressure Level (𝐿𝑝 dB re 20 µPa) [11]. The

surface of the inner hemisphere with radius 𝑟1 was

used to derivate the average sound pressure level,

which was used to calculate the sound power level

using the following simplified Equation (5):

Excerpt from the Proceedings of the 2015 COMSOL Conference in Curitiba

𝐿𝑤 = 𝐿𝑝 + 10 log10(𝑆) dB re 10−12 W, (5)

where 𝑆 = 2𝜋𝑟12 is the surface area of the inner

hemisphere, in m2.

The Insertion loss is the reduction of the sound

power level radiated by the enclosed source, and

thus, can be derived by:

𝐼𝐿 = 𝐿𝑤0 − 𝐿𝑤𝐸 , (6)

where 𝐿𝑤0 is the source power level and 𝐿𝑤𝐸 is

the sound power level of the enclosed source.

4. Results

This Section presents the experimental and

numerical results. Since the IL is derivate from the

sound power level, it makes sense to analyze at

first moment its results. In Figure 4, one can

compare the numerical and experimental results

for both studied situations. The numerical results

are obtained using Equation (5) and experimental

results by Equation (3).

Figure 4. Sound Power Level Results Comparison.

It can be seen in Figure 4, that the numerical

model showed very good agreement with

experimental curves. Despite the fact that the

numerical results for the enclosed source case

have lower magnitudes than the experimental

result, the curve shape is similar and predicts most

of the sound power peaks. The magnitude error

could be related to the boundary conditions and

material properties utilized to model the structural

domain. Also the sound absorption of the material

is not considered as its damping factor. Hence, the

structural modes of the enclosure are incorrectly

estimated, interfering on the fluid-structure

coupling results.

The numerical IL results are derived from

Equation (6) and the experimental results are

derived using the same idea, however deriving 𝐿𝑤

values by Equation (1) and Equation (3). The

results can be compared in Figure 5.

Figure 5. Insertion Loss Results Comparison.

Figure 5 shows that for frequencies lower than

100 Hz the enclosure is acoustic transparent,

meaning that it have no attenuation effects at this

range. In 660 Hz, a sudden decrease of IL is

notable, meaning that the radiated noise increase

and makes the enclosure a secondary source of

sound pressure. The same behavior is seen in

1080 Hz, 1300 Hz and 1674 Hz, where the IL

decrease is even higher and becomes almost equal

to zero.

Since there is an acoustic-structural coupling

between the internal cavity modes and the

external radiation domain, is important to

investigate the structural behavior of the

enclosure. This will allow to conclude about how

the structure resonances affects the IL curve.

Figure 6 shows the enclosure structural

displacement, which will give us an insight about

the enclosure’s structural modes.

Figure 6. Enclosure's Structural Displacement

(Surface Average).

In fact, by analyzing Figure 6, it can be seen

by looking at the most prominent peak that the

first mode of the structure is indeed at 660 Hz,

reinforcing the hypothesis that the IL decrease at

this frequency is due to a structural resonance.

By investigating the mode shapes in Figure 7

(a) and Figure 7 (b), is noticeable that bending

modes occurs at 230 Hz and 300 Hz, respectively.

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(a) (b)

(c) (d)

Figure 7. Surface Total Displacement [m] Color Plot.

Structural modes: (a) 230 Hz, (b) 300 Hz, (c) 660 Hz

and (d) 1080 Hz.

Figure 7 (c) shows that in 660 Hz the

enclosure mode occurs in all three dimensions,

thus increasing the sound radiation. Figure 7 (d)

shows how the resonance structural shape occurs

in 1080 Hz. Additionally, the directional pattern

presented in Figure 8 and Figure 9 allows a better

visualization of the directional acoustic radiation.

It can be seen that the radiation is higher in the

dimensions where the structural mode peaks takes

place. This proves that the modal resonances play

a important role on the radiated acoustic field.

Figure 8. Isosurface of the Total Acoustic Field [Pa]

at 660 Hz.

Figure 9. Isosurface of the Total Acoustic Field [Pa]

at 1080 Hz.

5. Conclusions

This study validated a multiphysical model to

estimate the Insertion Loss promoted by the

enclosure of a sound source. The model took into

account the source frequency dependent sound

power, which was obtained experimentally.

The IL results exposed in Figure 3 shows that

the numerical model provides a good estimate of

the IL behavior. However, the improvement of

this model still possible since more representative

boundary conditions and material properties

could lead to a better agreement between

experimental and numerical IL’ s curves.

Notwithstanding, the investigation about the

enclosure’s structural behavior showed that the

shell element was able to predict correctly the

resonance frequencies, thus validating the FE

model. This statement is supported by the fact that

the IL decrease at 660 Hz (Figure 5), for both

numerical and experimental cases, is in fact the

first structural mode of the enclosure, as seen in

Figure 6. The use of shell elements is justified

especially because of the low computational cost

required, in comparison to solid elements. In this

case, each frequency step took around 4 minutes

in a PC Core I7 processor with 16 GB RAM. Solid

elements could not be evaluated because of his

high RAM requirements.

Since approximated properties of the

enclosure’s wood was obtained from a database

[12], future works shall use optimization methods

to fit the numerical IL curve on the experimental

IL results. This optimization would have the

material properties as optimization variables.

Thus, would be possible to estimate real values of

material properties, which characterizes an

inverse acoustic problem. At the end of that work,

a reliable numerical model to estimate the IL

Excerpt from the Proceedings of the 2015 COMSOL Conference in Curitiba

promoted by the enclosure of a sound source

should be obtained, allowing its use as a powerful

tool for enclosure design and it’s IL optimization.

6. References

1. R. F. Barron, Industrial Noise Control, CRC

Press, 2002.

2. J. E. Blanks, Optimal Design of an Enclosure

for a Portable Generator, Master Thesis, Virginia

Tech (1997).

3. D. Oldham and S. Hillarby, The Acoustical

Performance of a Small Close Fitting

Enclousure, part 1: Theoretical Models. Journal

of Sound and Vibration, Volume 150(2),

261:281 (1991).

4. G. S. Birita. Sound Insulation of a Box.

PhD thesis, Technical University of Denmark,

Denmark, 2007.

5. Hambric, S. and Fahnline, J. Structural

Acoustics Tutorial—Part 2: Sound—Structure

Interaction. Acoustics Today, Volume 3(2),

9:27, (2007).

6. Sandberg, G. and Goransson, P. A symmetric

finite element formulation for acoustic fluid-

structure interaction analysis, Journal of Sound

and Vibration, Volume 123(3), 507:515, (1988).

7. Everstine, G and Henderson, F. Coupled finite

element/boundary element approach for fluid--

structure interaction, The Journal of the

Acoustical Society of America, Volume 87(5),

1938:1947, (1990).

8. ISO 3741:2010, Acoustics - Determination of

sound power levels and sound energy levels of

noise sources using sound pressure - Precision

methods for reverberation test rooms, 1999.

9. F.J. Fahy, “Measurement of Acoustic Intensity

Using the Cross-Spectral Density of two

Microphone Signals”, J. Acoust. Soc. Am.,

Volume 62, pp 1057-1059, 1977.

10. Atalla, N and Sgard, F. Finite Element and

Boundary Methods in Structural Acoustics and

Vibration, CRC Press, 2015.

11. Bies, David and Hansen, Colin. Engineering

Noise Control: Theory and Practice, CRC Press,

2009.

12. Institutos de Pesquisas Tecnológicas,

http://www.ipt.br/informacoes_madeiras/38.html

Excerpt from the Proceedings of the 2015 COMSOL Conference in Curitiba